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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>The <dfn class="terminology">general form of second order ODE</dfn> is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_1.html ./knowl/eq3_2.html">
\begin{equation*}
\frac{\textrm{d}^2 y}{\textrm{d} x^2}=f(x, y, \frac{\textrm{d} y}{\textrm{d} x}).
\end{equation*}
</div>
<p class="continuation">In particular, we consider</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_1.html ./knowl/eq3_2.html">
\begin{equation*}
f(x, y,  \frac{\textrm{d} y}{\textrm{d} x})=-p(x)  \frac{\textrm{d} y}{\textrm{d} x}-q(x) y+g(x),
\end{equation*}
</div>
<p class="continuation">i. e.,</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_1.html ./knowl/eq3_2.html">
\begin{equation}
\frac{\textrm{d}^2 y}{\textrm{d} x^2}+p(x) \frac{\textrm{d} y}{\textrm{d} x}+q(x) y=g(x).\tag{3.1.1}
\end{equation}
</div>
<p class="continuation">This is the <dfn class="terminology">general form of second order linear ODE</dfn>. Initial conditions</p>
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\begin{equation}
y(x_0)=y_0,\quad y^{\prime}(x_0)=y_1.\tag{3.1.2}
\end{equation}
</div>
<p class="continuation">Equations (<a href="" class="xref" data-knowl="./knowl/eq3_1.html" title="Equation 3.1.1">(3.1.1)</a>) together with (<a href="" class="xref" data-knowl="./knowl/eq3_2.html" title="Equation 3.1.2">(3.1.2)</a>) are called an <dfn class="terminology">initial value problem</dfn>.</p>
<span class="incontext"><a href="sec3_1.html#p-58" class="internal">in-context</a></span>
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